Search: id:A055830
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%I A055830
%S A055830 1,1,0,2,1,0,3,3,1,0,5,7,4,1,0,8,15,12,5,1,0,13,30,31,18,6,1,0,21,58,73,
%T A055830 54,25,7,1,0,34,109,162,145,85,33,8,1,0,55,201,344,361,255,125,42,9,1,
               0,
%U A055830 89,365,707,850,701,413,175,52,10,1,0,144
%N A055830 Triangle T read by rows: diagonal differences of triangle A037027.
%C A055830 Or, coefficients of a generalized Lucas-Pell polynomial read by rows. 
               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
%C A055830 Equals A046854(shifted) * Pascal's triangle; where A046854 is shifted 
               down one row and "1" inserted at (0,0). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 24 2008]
%F A055830 G.f.: (1-yz) / [1-y(1+y+z)].
%F A055830 T(i, j) = R(i-j, j), where R(0, 0)=1, R(0, j)=0 for j >= 1, R(1, j)=1 
               for j >= 0, R(i, j)=SUM{R(i-2, k)+R(i-1, k): k=0, 1, ..., j} for 
               i >= 1, j >= 1.
%F A055830 Sum_{k, 0<=k<=n}x^k*T(n,k)= A039834(n-2), A000012(n), A000045(n+1), A001333(n), 
               A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), 
               A015455(n), A015456(n), A015457(n) for x= -2,-1,0,1,2,3,4,5,6,7,8,
               9,10 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006
%F A055830 Sum_{k, 0<=k<=[n/2]}T(n-k,k)=A011782(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 22 2006
%F A055830 Triangle T(n,k), 0<=k<=n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA 
               [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined 
               in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 
               2006
%F A055830 T(n,0)= Fibonacci(n+1)=A000045(n+1) . Sum_{k, 0<=k<=n}T(n,k)=A001333(n) 
               . T(n,k)=0 if k>n or if k<0, T(0,0)=1, T(1,1)=0, T(n,k)=T(n-1,k-1)+T(n-1,
               k)+T(n-2,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 
               2006
%e A055830 1
%e A055830 1,0
%e A055830 2,1,0
%e A055830 3,3,1,0
%e A055830 5,7,4,1,0
%e A055830 8,15,12,5,1,0
%e A055830 13,30,31,18,6,1,0
%e A055830 21,58,73,54,25,7,1,0
%e A055830 34,109,162,145,85,33,8,1,0
%e A055830 55,201,344,361,255,125,42,9,1,0
%Y A055830 Left-hand columns include A000045, A023610.
%Y A055830 Right-hand columns include A055831, A055832, A055833, A055834, A055835, 
               A055836, A055837, A055838, A055839, A055840.
%Y A055830 Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)).
%Y A055830 Cf. A122075 (another version).
%Y A055830 A046854 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2008]
%Y A055830 Sequence in context: A100224 A089000 A107238 this_sequence A079123 A121548 
               A113020
%Y A055830 Adjacent sequences: A055827 A055828 A055829 this_sequence A055831 A055832 
               A055833
%K A055830 nonn,tabl
%O A055830 0,4
%A A055830 Clark Kimberling (ck6(AT)evansville.edu), May 28 2000
%E A055830 Edited by Ralf Stephan, Jan 12 2005

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